Jump to content

List of formulas in Riemannian geometry

From Wikipedia, the free encyclopedia

This is a list of formulas encountered in Riemannian geometry. Einstein notation is used throughout this article. This article uses the "analyst's" sign convention for Laplacians, except when noted otherwise.

Christoffel symbols, covariant derivative[edit]

In a smooth coordinate chart, the Christoffel symbols of the first kind are given by

and the Christoffel symbols of the second kind by

Here is the inverse matrix to the metric tensor . In other words,

and thus

is the dimension of the manifold.

Christoffel symbols satisfy the symmetry relations

or, respectively, ,


the second of which is equivalent to the torsion-freeness of the Levi-Civita connection.

The contracting relations on the Christoffel symbols are given by

and

where |g| is the absolute value of the determinant of the metric tensor . These are useful when dealing with divergences and Laplacians (see below).

The covariant derivative of a vector field with components is given by:

and similarly the covariant derivative of a -tensor field with components is given by:

For a -tensor field with components this becomes

and likewise for tensors with more indices.

The covariant derivative of a function (scalar) is just its usual differential:

Because the Levi-Civita connection is metric-compatible, the covariant derivatives of metrics vanish,

as well as the covariant derivatives of the metric's determinant (and volume element)

The geodesic starting at the origin with initial speed has Taylor expansion in the chart:

Curvature tensors[edit]

Definitions[edit]

(3,1) Riemann curvature tensor[edit]

(3,1) Riemann curvature tensor[edit]

Ricci curvature[edit]

Scalar curvature[edit]

Traceless Ricci tensor[edit]

(4,0) Riemann curvature tensor[edit]

(4,0) Weyl tensor[edit]

Einstein tensor[edit]

Identities[edit]

Basic symmetries[edit]

The Weyl tensor has the same basic symmetries as the Riemann tensor, but its 'analogue' of the Ricci tensor is zero:

The Ricci tensor, the Einstein tensor, and the traceless Ricci tensor are symmetric 2-tensors:

First Bianchi identity[edit]

Second Bianchi identity[edit]

Contracted second Bianchi identity[edit]

Twice-contracted second Bianchi identity[edit]

Equivalently:

Ricci identity[edit]

If is a vector field then

which is just the definition of the Riemann tensor. If is a one-form then

More generally, if is a (0,k)-tensor field then

Remarks[edit]

A classical result says that if and only if is locally conformally flat, i.e. if and only if can be covered by smooth coordinate charts relative to which the metric tensor is of the form for some function on the chart.

Gradient, divergence, Laplace–Beltrami operator[edit]

The gradient of a function is obtained by raising the index of the differential , whose components are given by:

The divergence of a vector field with components is

The Laplace–Beltrami operator acting on a function is given by the divergence of the gradient:

The divergence of an antisymmetric tensor field of type simplifies to

The Hessian of a map is given by

Kulkarni–Nomizu product[edit]

The Kulkarni–Nomizu product is an important tool for constructing new tensors from existing tensors on a Riemannian manifold. Let and be symmetric covariant 2-tensors. In coordinates,

Then we can multiply these in a sense to get a new covariant 4-tensor, which is often denoted . The defining formula is

Clearly, the product satisfies

In an inertial frame[edit]

An orthonormal inertial frame is a coordinate chart such that, at the origin, one has the relations and (but these may not hold at other points in the frame). These coordinates are also called normal coordinates. In such a frame, the expression for several operators is simpler. Note that the formulae given below are valid at the origin of the frame only.

Conformal change[edit]

Let be a Riemannian or pseudo-Riemanniann metric on a smooth manifold , and a smooth real-valued function on . Then

is also a Riemannian metric on . We say that is (pointwise) conformal to . Evidently, conformality of metrics is an equivalence relation. Here are some formulas for conformal changes in tensors associated with the metric. (Quantities marked with a tilde will be associated with , while those unmarked with such will be associated with .)

Levi-Civita connection[edit]

(4,0) Riemann curvature tensor[edit]

  • where

Using the Kulkarni–Nomizu product:

Ricci tensor[edit]

Scalar curvature[edit]

  • if this can be written

Traceless Ricci tensor[edit]

(3,1) Weyl curvature[edit]

  • for any vector fields

Volume form[edit]

Hodge operator on p-forms[edit]

Codifferential on p-forms[edit]

Laplacian on functions[edit]

Hodge Laplacian on p-forms[edit]

The "geometer's" sign convention is used for the Hodge Laplacian here. In particular it has the opposite sign on functions as the usual Laplacian.

Second fundamental form of an immersion[edit]

Suppose is Riemannian and is a twice-differentiable immersion. Recall that the second fundamental form is, for each a symmetric bilinear map which is valued in the -orthogonal linear subspace to Then

  • for all

Here denotes the -orthogonal projection of onto the -orthogonal linear subspace to

Mean curvature of an immersion[edit]

In the same setting as above (and suppose has dimension ), recall that the mean curvature vector is for each an element defined as the -trace of the second fundamental form. Then

Note that this transformation formula is for the mean curvature vector, and the formula for the mean curvature in the hypersurface case is

where is a (local) normal vector field.

Variation formulas[edit]

Let be a smooth manifold and let be a one-parameter family of Riemannian or pseudo-Riemannian metrics. Suppose that it is a differentiable family in the sense that for any smooth coordinate chart, the derivatives exist and are themselves as differentiable as necessary for the following expressions to make sense. is a one-parameter family of symmetric 2-tensor fields.

Principal symbol[edit]

The variation formula computations above define the principal symbol of the mapping which sends a pseudo-Riemannian metric to its Riemann tensor, Ricci tensor, or scalar curvature.

  • The principal symbol of the map assigns to each a map from the space of symmetric (0,2)-tensors on to the space of (0,4)-tensors on given by
  • The principal symbol of the map assigns to each an endomorphism of the space of symmetric 2-tensors on given by
  • The principal symbol of the map assigns to each an element of the dual space to the vector space of symmetric 2-tensors on by

See also[edit]

Notes[edit]

References[edit]

  • Arthur L. Besse. "Einstein manifolds." Ergebnisse der Mathematik und ihrer Grenzgebiete (3) [Results in Mathematics and Related Areas (3)], 10. Springer-Verlag, Berlin, 1987. xii+510 pp. ISBN 3-540-15279-2